### Nuprl Lemma : rep-sub-sheaf_wf

`∀[C:SmallCategory]. ∀[X:cat-ob(C)]. ∀[P:U:cat-ob(C) ⟶ (cat-arrow(C) U X) ⟶ ℙ].`
`  rep-sub-sheaf(C;X;P) ∈ Functor(op-cat(C);TypeCat) `
`  supposing ∀A,B:cat-ob(C). ∀g:cat-arrow(C) A B. ∀b:{b:cat-arrow(C) B X| P B b} .  (P A (cat-comp(C) A B X g b))`

Proof

Definitions occuring in Statement :  rep-sub-sheaf: `rep-sub-sheaf(C;X;P)` type-cat: `TypeCat` op-cat: `op-cat(C)` cat-functor: `Functor(C1;C2)` cat-comp: `cat-comp(C)` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` rep-sub-sheaf: `rep-sub-sheaf(C;X;P)` cat-functor: `Functor(C1;C2)` small-category: `SmallCategory` cat-comp: `cat-comp(C)` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` pi1: `fst(t)` pi2: `snd(t)` type-cat: `TypeCat` op-cat: `op-cat(C)` spreadn: spread4 and: `P ∧ Q` subtype_rel: `A ⊆r B` cat-id: `cat-id(C)` compose: `f o g` cand: `A c∧ B` all: `∀x:A. B[x]` squash: `↓T` prop: `ℙ` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  equal_wf squash_wf true_wf iff_weakening_equal set_wf all_wf cat-ob_wf op-cat_wf cat-arrow_wf type-cat_wf cat-id_wf cat-comp_wf small-category_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut dependent_set_memberEquality sqequalHypSubstitution setElimination thin rename productElimination sqequalRule dependent_pairEquality lambdaEquality setEquality applyEquality functionExtensionality hypothesisEquality cumulativity because_Cache hypothesis functionEquality lambdaFormation independent_pairFormation imageElimination extract_by_obid isectElimination equalityTransitivity equalitySymmetry universeEquality dependent_functionElimination natural_numberEquality imageMemberEquality baseClosed independent_isectElimination independent_functionElimination productEquality instantiate axiomEquality isect_memberEquality

Latex:
\mforall{}[C:SmallCategory].  \mforall{}[X:cat-ob(C)].  \mforall{}[P:U:cat-ob(C)  {}\mrightarrow{}  (cat-arrow(C)  U  X)  {}\mrightarrow{}  \mBbbP{}].
rep-sub-sheaf(C;X;P)  \mmember{}  Functor(op-cat(C);TypeCat)
supposing  \mforall{}A,B:cat-ob(C).  \mforall{}g:cat-arrow(C)  A  B.  \mforall{}b:\{b:cat-arrow(C)  B  X|  P  B  b\}  .
(P  A  (cat-comp(C)  A  B  X  g  b))

Date html generated: 2017_10_05-AM-00_47_10
Last ObjectModification: 2017_07_28-AM-09_19_34

Theory : small!categories

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